If $f'' \ge 0$, $\int_0^2 f(x)dx \ge 2f(1)$

By a function form of Jensen's inequality, it holds that $\frac{1}{2} \int_0^2f(x)dx \geq f(\frac{1}{2}\int_0^2xdx) = f(1)$. (See https://en.wikipedia.org/wiki/Jensen%27s_inequality)